Is there a list of all connected T_0-spaces with 5 points?
Peter Jipsen has granted me permission to post static images of his posets.
4-element connected posets (10):
5-element connected posets (44):
Since the number (139) of distinct 5-element topological spaces (up to homeomorphism) is manageably small and there don't seem to be any lists posted anywhere, I am posting one here. The first (second) row lists smallest closed (open) supersets of singletons and the rows that follow list all nontrivial open sets.
1 T_0
a b c d e
-----
a b c d e
-----
a b c d e
ab ac ad ae bc bd be cd ce de
abc abd abe acd ace ade bcd bce bde cde
abcd abce abde acde bcde
2 T_0
ae b c d e
-----
a b c d ae
-----
a b c d
ab ac ad ae bc bd cd
abc abd abe acd ace ade bcd
abcd abce abde acde
3 T_0
ae be c d e
-----
a b c d abe
-----
a b c d
ab ac ad bc bd cd
abc abd abe acd bcd
abcd abce abde
4 T_0
ae be ce d e
-----
a b c d abce
-----
a b c d
ab ac ad bc bd cd
abc abd acd bcd
abcd abce
5 connected T_0, poset (4,2)
ae be ce de e
-----
a b c d abcde
-----
a b c d
ab ac ad bc bd cd
abc abd acd bcd
abcd
6 T_0
ade b c d e
-----
a b c ad ae
-----
a b c
ab ac ad ae bc
abc abd abe acd ace ade
abcd abce abde acde
7 T_0
ad be c d e
-----
a b c ad be
-----
a b c
ab ac ad bc be
abc abd abe acd bce
abcd abce abde
8 T_0
ade be c d e
-----
a b c ad abe
-----
a b c
ab ac ad bc
abc abd abe acd
abcd abce abde
9 T_0
ade b c de e
-----
a b c ad ade
-----
a b c
ab ac ad bc
abc abd acd ade
abcd abde acde
10
a b c de de
-----
a b c de de
-----
a b c
ab ac bc de
abc ade bde cde
abde acde bcde
11 T_0
ad be ce d e
-----
a b c ad bce
-----
a b c
ab ac ad bc
abc abd acd bce
abcd abce
12 connected T_0, poset (1,9)
ade be ce d e
-----
a b c ad abce
-----
a b c
ab ac ad bc
abc abd acd
abcd abce
13 T_0
ade be c de e
-----
a b c ad abde
-----
a b c
ab ac ad bc
abc abd acd
abcd abde
14 connected T_0, poset (4,3)
ade be ce de e
-----
a b c ad abcde
-----
a b c
ab ac ad bc
abc abd acd
abcd
15 T_0
ade bde c d e
-----
a b c abd abe
-----
a b c
ab ac bc
abc abd abe
abcd abce abde
16 connected T_0, poset (2,6)
ade bd ce d e
-----
a b c abd ace
-----
a b c
ab ac bc
abc abd ace
abcd abce
17 connected T_0, poset (2,9)
ade bde ce d e
-----
a b c abd abce
-----
a b c
ab ac bc
abc abd
abcd abce
18 T_0
ade bde c de e
-----
a b c abd abde
-----
a b c
ab ac bc
abc abd
abcd abde
19
ade b c de de
-----
a b c ade ade
-----
a b c
ab ac bc
abc ade
abde acde
20 connected T_0, poset (4,6)
ade bde ce de e
-----
a b c abd abcde
-----
a b c
ab ac bc
abc abd
abcd
21 connected T_0, poset (3,6)
ade bde cde d e
-----
a b c abcd abce
-----
a b c
ab ac bc
abc
abcd abce
22 connected T_0, poset (5,4)
ade bde cde de e
-----
a b c abcd abcde
-----
a b c
ab ac bc
abc
abcd
23
ade bde c de de
-----
a b c abde abde
-----
a b c
ab ac bc
abc
abde
24 connected
ade bde cde de de
-----
a b c abcde abcde
-----
a b c
ab ac bc
abc
25 T_0
acde b c d e
-----
a b ac ad ae
-----
a b
ab ac ad ae
abc abd abe acd ace ade
abcd abce abde acde
26 T_0
acd be c d e
-----
a b ac ad be
-----
a b
ab ac ad be
abc abd abe acd
abcd abce abde
27 connected T_0, poset (1,2)
acde be c d e
-----
a b ac ad abe
-----
a b
ab ac ad
abc abd abe acd
abcd abce abde
28 T_0
acde b ce d e
-----
a b ac ad ace
-----
a b
ab ac ad
abc abd acd ace
abcd abce acde
29 connected T_0, poset (1,4)
ace bde c d e
-----
a b ac bd abe
-----
a b
ab ac bd
abc abd abe
abcd abce abde
30 connected T_0, poset (2,1)
acde be ce d e
-----
a b ac ad abce
-----
a b
ab ac ad
abc abd acd
abcd abce
31 T_0
acde b ce de e
-----
a b ac ad acde
-----
a b
ab ac ad
abc abd acd
abcd acde
32 T_0
ace bd ce d e
-----
a b ac bd ace
-----
a b
ab ac bd
abc abd ace
abcd abce
33
ac b c de de
-----
a b ac de de
-----
a b
ab ac de
abc ade bde
abde acde
34 connected T_0, poset (4,4)
acde be ce de e
-----
a b ac ad abcde
-----
a b
ab ac ad
abc abd acd
abcd
35 connected T_0, poset (2,3)
ace bde ce d e
-----
a b ac bd abce
-----
a b
ab ac bd
abc abd
abcd abce
36 connected T_0, poset (4,8)
ace bde ce de e
-----
a b ac bd abcde
-----
a b
ab ac bd
abc abd
abcd
37 connected T_0, poset (1,5)
acde bde c d e
-----
a b ac abd abe
-----
a b
ab ac
abc abd abe
abcd abce abde
38 T_0
acde b cde d e
-----
a b ac acd ace
-----
a b
ab ac
abc acd ace
abcd abce acde
39
a b cde cde e
-----
a b cd cd cde
-----
a b
ab cd
acd bcd cde
abcd acde bcde
40 connected T_0, poset (2,7)
acde bd ce d e
-----
a b ac abd ace
-----
a b
ab ac
abc abd ace
abcd abce
41
ac bc c de de
-----
a b abc de de
-----
a b
ab de
abc ade bde
abde
42 connected T_0, poset (3,2)
acde bde ce d e
-----
a b ac abd abce
-----
a b
ab ac
abc abd
abcd abce
43 connected T_0, poset (2,4)
acde bde c de e
-----
a b ac abd abde
-----
a b
ab ac
abc abd
abcd abde
44 connected T_0, poset (3,1)
acde be cde d e
-----
a b ac acd abce
-----
a b
ab ac
abc acd
abcd abce
45 T_0
acde b cde de e
-----
a b ac acd acde
-----
a b
ab ac
abc acd
abcd acde
46
acde b c de de
-----
a b ac ade ade
-----
a b
ab ac
abc ade
abde acde
47
ae b cde cde e
-----
a b cd cd acde
-----
a b
ab cd
acd bcd
abcd acde
48 connected T_0, poset (4,7)
acde bde ce de e
-----
a b ac abd abcde
-----
a b
ab ac
abc abd
abcd
49 connected T_0, poset (4,9)
acde be cde de e
-----
a b ac acd abcde
-----
a b
ab ac
abc acd
abcd
50
ac bde c de de
-----
a b ac bde bde
-----
a b
ab ac
abc bde
abde
51 connected
ae be cde cde e
-----
a b cd cd abcde
-----
a b
ab cd
acd bcd
abcd
52 connected T_0, poset (3,7)
acde bde cde d e
-----
a b ac abcd abce
-----
a b
ab ac
abc
abcd abce
53 connected T_0, poset (5,5)
acde bde cde de e
-----
a b ac abcd abcde
-----
a b
ab ac
abc
abcd
54 connected
acde bde c de de
-----
a b ac abde abde
-----
a b
ab ac
abc
abde
55
acde b cde de de
-----
a b ac acde acde
-----
a b
ab ac
abc
acde
56 connected
acde bde cde de de
-----
a b ac abcde abcde
-----
a b
ab ac
abc
57 connected T_0, poset (1,7)
acde bcde c d e
-----
a b abc abd abe
-----
a b
ab
abc abd abe
abcd abce abde
58 connected T_0, poset (3,4)
acde bcde ce d e
-----
a b abc abd abce
-----
a b
ab
abc abd
abcd abce
59 connected T_0, poset (5,2)
acde bcde ce de e
-----
a b abc abd abcde
-----
a b
ab
abc abd
abcd
60 connected
acde bc c de de
-----
a b abc ade ade
-----
a b
ab
abc ade
abde
61 connected T_0, poset (3,9)
acde bcde cde d e
-----
a b abc abcd abce
-----
a b
ab
abc
abcd abce
62
acde b cde cde e
-----
a b acd acd acde
-----
a b
ab
acd
abcd acde
63
a b cde cde cde
-----
a b cde cde cde
-----
a b
ab
cde
acde bcde
64 connected T_0, poset (5,7)
acde bcde cde de e
-----
a b abc abcd abcde
-----
a b
ab
abc
abcd
65 connected
acde bcde c de de
-----
a b abc abde abde
-----
a b
ab
abc
abde
66 connected
acde be cde cde e
-----
a b acd acd abcde
-----
a b
ab
acd
abcd
67 connected
acde bcde cde de de
-----
a b abc abcde abcde
-----
a b
ab
abc
68 connected
acde bcde cde cde e
-----
a b abcd abcd abcde
-----
a b
ab
abcd
69
acde b cde cde cde
-----
a b acde acde acde
-----
a b
ab
acde
70 connected
acde bcde cde cde cde
-----
a b abcde abcde abcde
-----
a b
ab
71 connected T_0, poset (1,1)
abcde b c d e
-----
a ab ac ad ae
-----
a
ab ac ad ae
abc abd abe acd ace ade
abcd abce abde acde
72 connected T_0, poset (1,3)
abcde be c d e
-----
a ab ac ad abe
-----
a
ab ac ad
abc abd abe acd
abcd abce abde
73 connected T_0, poset (2,2)
abcde be ce d e
-----
a ab ac ad abce
-----
a
ab ac ad
abc abd acd
abcd abce
74 connected T_0, poset (4,5)
abcde be ce de e
-----
a ab ac ad abcde
-----
a
ab ac ad
abc abd acd
abcd
75
abc b c de de
-----
a ab ac de de
-----
a
ab ac de
abc ade
abde acde
76 connected T_0, poset (1,6)
abcde bde c d e
-----
a ab ac abd abe
-----
a
ab ac
abc abd abe
abcd abce abde
77 connected T_0, poset (2,8)
abcde bd ce d e
-----
a ab ac abd ace
-----
a
ab ac
abc abd ace
abcd abce
78 connected T_0, poset (3,3)
abcde bde ce d e
-----
a ab ac abd abce
-----
a
ab ac
abc abd
abcd abce
79 connected T_0, poset (2,5)
abcde bde c de e
-----
a ab ac abd abde
-----
a
ab ac
abc abd
abcd abde
80 connected
abcde b c de de
-----
a ab ac ade ade
-----
a
ab ac
abc ade
abde acde
81
ab b cde cde e
-----
a ab cd cd cde
-----
a
ab cd
acd cde
abcd acde
82 connected T_0, poset (5,1)
abcde bde ce de e
-----
a ab ac abd abcde
-----
a
ab ac
abc abd
abcd
83
abc bc c de de
-----
a ab abc de de
-----
a
ab de
abc ade
abde
84
a bc bc de de
-----
a bc bc de de
-----
a
bc de
abc ade
bcde
85 connected T_0, poset (3,8)
abcde bde cde d e
-----
a ab ac abcd abce
-----
a
ab ac
abc
abcd abce
86 connected
abe b cde cde e
-----
a ab cd cd acde
-----
a
ab cd
acd
abcd acde
87 connected T_0, poset (5,6)
abcde bde cde de e
-----
a ab ac abcd abcde
-----
a
ab ac
abc
abcd
88 connected
abcde bde c de de
-----
a ab ac abde abde
-----
a
ab ac
abc
abde
89 connected
abe be cde cde e
-----
a ab cd cd abcde
-----
a
ab cd
acd
abcd
90 connected
abcde bde cde de de
-----
a ab ac abcde abcde
-----
a
ab ac
abc
91 connected T_0, poset (1,8)
abcde bcde c d e
-----
a ab abc abd abe
-----
a
ab
abc abd abe
abcd abce abde
92
a bcde bcde d e
-----
a bc bc bcd bce
-----
a
bc
abc bcd bce
abcd abce bcde
93 connected T_0, poset (3,5)
abcde bcde ce d e
-----
a ab abc abd abce
-----
a
ab
abc abd
abcd abce
94 connected
ae bcde bcde d e
-----
a bc bc bcd abce
-----
a
bc
abc bcd
abcd abce
95
a bcde bcde de e
-----
a bc bc bcd bcde
-----
a
bc
abc bcd
abcd bcde
96 connected T_0, poset (5,3)
abcde bcde ce de e
-----
a ab abc abd abcde
-----
a
ab
abc abd
abcd
97 connected
abcde bc c de de
-----
a ab abc ade ade
-----
a
ab
abc ade
abde
98 connected
ae bcde bcde de e
-----
a bc bc bcd abcde
-----
a
bc
abc bcd
abcd
99
ade bc bc de de
-----
a bc bc ade ade
-----
a
bc
abc ade
100 connected T_0, poset (4,1)
abcde bcde cde d e
-----
a ab abc abcd abce
-----
a
ab
abc
abcd abce
101 connected
abcde b cde cde e
-----
a ab acd acd acde
-----
a
ab
acd
abcd acde
102 connected
ade bcde bcde d e
-----
a bc bc abcd abce
-----
a
bc
abc
abcd abce
103 connected T_0, poset (5,8)
abcde bcde cde de e
-----
a ab abc abcd abcde
-----
a
ab
abc
abcd
104 connected
abcde bcde c de de
-----
a ab abc abde abde
-----
a
ab
abc
abde
105 connected
abcde be cde cde e
-----
a ab acd acd abcde
-----
a
ab
acd
abcd
106
ab b cde cde cde
-----
a ab cde cde cde
-----
a
ab
cde
acde
107 connected
ade bcde bcde de e
-----
a bc bc abcd abcde
-----
a
bc
abc
abcd
108
a bcde bcde de de
-----
a bc bc bcde bcde
-----
a
bc
abc
bcde
109 connected
abcde bcde cde de de
-----
a ab abc abcde abcde
-----
a
ab
abc
110 connected
ade bcde bcde de de
-----
a bc bc abcde abcde
-----
a
bc
abc
111 connected
abcde bcde cde cde e
-----
a ab abcd abcd abcde
-----
a
ab
abcd
112 connected
abcde b cde cde cde
-----
a ab acde acde acde
-----
a
ab
acde
113 connected
abcde bcde cde cde cde
-----
a ab abcde abcde abcde
-----
a
ab
114 connected
abcde bc bc de de
-----
a abc abc ade ade
-----
a
abc ade
115 connected
abcde bcde bcde d e
-----
a abc abc abcd abce
-----
a
abc
abcd abce
116
a bcde bcde bcde e
-----
a bcd bcd bcd bcde
-----
a
bcd
abcd bcde
117 connected
abcde bcde bcde de e
-----
a abc abc abcd abcde
-----
a
abc
abcd
118 connected
ae bcde bcde bcde e
-----
a bcd bcd bcd abcde
-----
a
bcd
abcd
119 connected
abcde bcde bcde de de
-----
a abc abc abcde abcde
-----
a
abc
120 connected
abcde bcde bcde bcde e
-----
a abcd abcd abcd abcde
-----
a
abcd
121
a bcde bcde bcde bcde
-----
a bcde bcde bcde bcde
-----
a
bcde
122 connected
abcde bcde bcde bcde bcde
-----
a abcde abcde abcde abcde
-----
a
123
abc abc c de de
-----
ab ab abc de de
-----
ab de
abc
abde
124 connected
abe abe cde cde e
-----
ab ab cd cd abcde
-----
ab cd
abcd
125 connected
abcde abcde c d e
-----
ab ab abc abd abe
-----
ab
abc abd abe
abcd abce abde
126 connected
abcde abcde ce d e
-----
ab ab abc abd abce
-----
ab
abc abd
abcd abce
127 connected
abcde abcde ce de e
-----
ab ab abc abd abcde
-----
ab
abc abd
abcd
128 connected
abcde abcde cde d e
-----
ab ab abc abcd abce
-----
ab
abc
abcd abce
129 connected
abcde abcde cde de e
-----
ab ab abc abcd abcde
-----
ab
abc
abcd
130 connected
abcde abcde c de de
-----
ab ab abc abde abde
-----
ab
abc
abde
131 connected
abcde abcde cde de de
-----
ab ab abc abcde abcde
-----
ab
abc
132
ab ab cde cde cde
-----
ab ab cde cde cde
-----
ab
cde
133 connected
abcde abcde cde cde e
-----
ab ab abcd abcd abcde
-----
ab
abcd
134 connected
abcde abcde cde cde cde
-----
ab ab abcde abcde abcde
-----
ab
135 connected
abcde abcde abcde d e
-----
abc abc abc abcd abce
-----
abc
abcd abce
136 connected
abcde abcde abcde de e
-----
abc abc abc abcd abcde
-----
abc
abcd
137 connected
abcde abcde abcde de de
-----
abc abc abc abcde abcde
-----
abc
138 connected
abcde abcde abcde abcde e
-----
abcd abcd abcd abcd abcde
-----
abcd
139 connected
abcde abcde abcde abcde abcde
-----
abcde abcde abcde abcde abcde
-----
trivial topology
There is a Java applet that displays all 5-element connected posets at http://www1.chapman.edu/~jipsen/gap/posets.html.
At the online encyclopedia of integer sequences we find, when we type T_0 topologies several hits. Sequence A028856 is the sequence of homeomorphism classes of T_0 topologies, and A028858 has all connected ones (308 topologies of which 235 connected, on 5 points). No explicit list of spaces, though, but some literature references that might help.